Now we are ready for physically interpreting equation 3.7,
which can be rearranged as:

=

where
.
Recalling Parseval's theorem which says
,
equation 3.17 can be compactly represented
by the following formula:

=

Here the forward modeled wavefield and backward modeled residuals are given by:

=

=

(3.15)

Equation 3.18 says the backpropagated residual wavefield
is crosscorrelated with the forward propagated wavefield
to give the slowness update.

To illustrate the physical meaning of equation 3.19, the
top picture in Figure 3.8
depicts the wavefronts emanating from a source in a homogeneous medium with an
embedded point scatterer. The combined direct and scattered wavefields can be
represented as
.
If the homogeneous
medium is used as the model velocity then the middle figure depicts the forward
propagated field, where no scattering from the point scatterer is extant. This field
is represented by
.
The bottom figure represents the backpropagated
scattered field (i.e., the residual field
)
denoted
by
.
The operation of crosscorrelation at zero lag is equivalent to multiplying each snapshot
in the middle figure by the corresponding snapshot in the bottom figure to give
a product snapshot, and these product snapshots are then summed over the time to give the image panel.
This image panel is zero everywhere except at the location of the scatterer
because that is the only location where the downgoing direct wavefront is coincident in
both space and time.

Figure 3.8:
Snapshots of wavefronts for a point source located at the origin, and a
buried scatterer denoted by *. Top figure depicts the total wavefield snapshots, middle figure
depicts the forward propagated wavefield in a homogeneous medium,
and bottom figure depicts the backpropagated scattered field.

The update formula
for many different inversion algorithms reduces to
"migrating" data residuals. The data residuals
might be associated with the traveltimes, autocorrelograms, phases, or amplitudes.